HFCC Math Lab Intermediate Algebra - 8 ADDITION AND SUBTRATION OF RATIONAL EXPRESSIONS Adding or subtracting two rational expressions require the rational expressions to have the same denominator. Example : Adding two rational expressions with the common denominator x+ 5 x x = 5 x 8 = x When the denominators of two rational expressions are not the same we write equivalent rational expression for both or one of the rational expressions so that they both have same denominator. To find the equivalent rational expressions we multiply the numerator and denominator of a rational function with same non-zero factor. Example : A list of equivalent rational expressions to x x 0 0(x ) => 0 0( ) x x x => x(x ) x( ) assuming x 0 x (x (x => (x )( x ( )( x assuming x 0 x 0 x (x 5) 0 x (x 5) 0 x (x 5)(x ) 0 ( 5)(5 ) => x x x assuming 0 x (x 5) 0 Remark: Try to reason out why every rational expression in the list above is equivalent to x We write equivalent fractions so that the two rational expressions have the same denominator. The best choice for the denominator is the Least Common Denominator (LCD). See Intermediate Algebra-8 hand out to learn how to find LCD Remark: Try to reason why the above statement makes sense. (Hint: Least means smallest multiple so a polynomial. A smallest multiple of a polynomial has the least degree and yet can be written as a product of both the polynomials under consideration.) Revised /09
In all the examples and exercises assume that the denominator is not zero. To add or subtract two or more rational expressions you can follow this procedure:. Factor each denominator completely, showing repeated factors as powers. If you have the - factor move the negative sign to the numerator so that the denominator factor is does not contain the - factor.. If the denominators are the same go to next step otherwise a) Find the LCD of the given expressions. b) Write equivalent fractions for each expression so that all the resulting fractions have the same denominator.. Combine numerator and eliminate parentheses in the numerator if any 4. Combine like terms. 5. If possible simplify by factoring the numerator and reducing. Remark: When subtracting two rational expressions make the necessary sign changes to the expression that is begin subtracted. Example : Add x 5 x 6x 6x Solution: Use the above procedure to add the two expressions x (x ) (x ) Same denominator ( x) (5 x) (x ) x (x ) 4x (x ) (x ) (x ) Step : Factor each denominator completely Step : Skip Step : Combine numerator Eliminate the parenthesis in the numerator Step 4: Combine like terms Step 5: Simplify if possible Revised /09
x x Example 4: Subtract 4x 6 4x 6 Solution: Use the above procedures to add the two expressions x x (x (x Same Denominator ( x) ( x (x x x (x x (x (x (x Step : Factor each denominator completely Step : Skip Step : Combine the numerators Eliminate the parenthesis in the numerator Pay attention to the negative sign when subtracting Step 4: Combine like terms Step 5: Simplify if possible Example 5: Add x x x 4 Solution: Use the above procedure to add the two expressions ( x )( x ) ( x 4)( x ) Denominator not same LCD ( x )( x )( x 4) ( x 4) ( x ) ( x )( x ) ( x 4) ( x 4)( x ) ( x ) ( x 4) ( x ) ( x )( x )( x 4) x 4 x ( x )( x )( x 4) Step : Factor each denominator completely Step : a) Find the LCD b) For each fraction write the equivalent fraction that has LCD as its denominator Step : Combine numerator Eliminate the parenthesis in the numerator x ( x )( x )( x 4) Step 4: Combine like terms Revised /09
( x ) ( x ) ( x ) ( x 4) ( x )( x 4) Step 5: If possible simplify x x Example 6: Subtract x 6 x 8x 5 Solution: Use the above procedure to add the two expressions x x ( x )( x ( x ( x 5) Denominators are not the same LCD ( x )( x ( x 5) ( x ) ( x 5) (x ) ( x ) ( x )( x ( x 5) ( x ( x 5) ( x ) Step : Factor each denominator completely Step : a) Find the LCD b) For each fraction write the equivalent fraction that has LCD as its denominator ( x )( x 5) (x )( x ) ( x )( x ( x 5) ( x 6x 5) (x x ) ( x )( x ( x 5) x x x x 6 5 ( x )( x ( x 5) Step : Combine numerator Eliminate the parenthesis in the numerator Pay attention to the negative sign when subtracting x x 7 ( x )( x ( x 5) Step 4: Combine like terms x x 7 ( x )( x ( x 5) Step 5: If possible simplify 5 Example 7: Add 4a b 6ab Solution: Use the above procedure to add the two expressions 5 a b a b Denominator are not the same LCD a b a b Revised /09 4 Step : Factor each denominator completely Step : a) Find the LCD b) For each fraction write the equivalent fraction that has LCD as its denominator
( b) 5 ( a) ( a b ) ( b) ( a b ) ( a) ( b) 5( a) ab 9b 0a ab 9b 0a ab Step : Combine numerators Eliminate the parenthesis in the numerator Step 4: Combine like terms if possible Step 5: Not possible of simplify the answer Exercises: Perform the indicated operation and then simplify your answer (Hint: follow the procedure suggested above). x y 4xy 6xy. 4 4 x y 4xy. x x 4 4. 4 x 5 x 5. a a 9 7. x x x 6 9. x x x 6. b 6 b 4 8. x x x 6 0. x 5 x 5. x x 4 4 x. x x x 5 5 x Solutions to the odd-numbered exercises and answers to the even- numbered exercises: 6 x ( x).. or 4 4 x y 4xy 6xy 4xy 4xy LCD = x y ()(6 y ) ()( x) ()( xy ) ( x y)(6 y ) (4 xy )( xy) (6 xy)( xy ) 6y x xy xy Revised /09 5
. x x 4 LCD = ( x ( x 4) ()( x 4) (( x ( x ( x 4) ( x 4)( x ()( x 4) ( x ( x ( x 4) (x 8) (x 9) ( x ( x 4) x 8 x 9 ( x ( x 4) x ( x ( x 4) 4. x 6 ( x or ( x 5)( x ) ( x 5)( x ) 5. a a 9 a ( a ( a LCD = ( a ( a ()( a ( a ( a ( a ( a ()( a ( a ( a a 6 ( a ( a a 6 ( a ( a ( a ( a ( a ( a 6. b 0 ( b 4)( b 4) 7. x x x 6 ( x )( x ) ( x )( x Revised /09 6 8. x or ( x ( x 4)( x 4) ( x ) ( x ( x 4)( x 4)
LCD = ( x )( x )( x ( x ( x ) ( x )( x )( x ( x )( x ( x ) ( x ( x ) ( x )( x )( x (x 6) ( x ) ( x )( x )( x x 6 x ( x )( x )( x x 5 ( x )( x )( x x 5 ( x )( x )( x 9. x x x ( x )( x ) ( x )( x ) LCD = ( x )( x )( x ) ()( x ) (( x ) ( x )( x )( x ) ( x )( x )( x ) ()( x ) (( x ) ( x )( x )( x ) (x 4) (x ( x )( x )( x ) x 4 x ( x )( x )( x ) ( x )( x )( x ) 0. x ( x 5)( x 5). x 4 x 4 x. x x 5 or x 5 x Revised /09 7
(x ( x 4) ( x 4) (x ( x 4) ( x 4) (x ( x 4) ( x 4) Common Denominator (x ) ( x 4) x ( x 4) x ( x 4) Answer x x or x 4 4 x You can get additional instructions and practice for solving these problems by going to the following websites: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_t ut0_addrat.htm#lcd This website provides video demonstration and step-bystep instruction on how to add two rational expressions. http://www.regentsprep.org/regents/math/algtrig/ato/addfrac.htm Student friendly notes on adding and subtracting rational functions http://www.purplemath.com/modules/rtnladd.htm This website has step-by-step instruction on how to add rational numbers and rational expressions. Make sure to visit all the three pages of notes by clicking on the numbers at the bottom of the web page. Revised /09 8